Timeline for Convergence of heat flow on non-compact manifolds?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 20, 2023 at 16:24 | comment | added | Student | Thanks a lot for your answer, this has been very helpful! | |
| Apr 20, 2023 at 16:20 | vote | accept | Student | ||
| Apr 20, 2023 at 13:54 | comment | added | Willie Wong | For your question that is completely unimportant. The proof that @MaoWao provided works equally well on compact manifolds, possibly with positive curvature. But on compacts since the spectrum is discrete you will have spectral gap and the convergence is exponential. | |
| Apr 20, 2023 at 13:50 | comment | added | Overflowian | By the way, you didn't use the extra hypothesis of non-positive curvature, didn't you? | |
| Apr 20, 2023 at 12:55 | comment | added | MaoWao | Yes, essential self-adjointness is stable under additive bounded (self-adjoint) perturbations. | |
| Apr 20, 2023 at 12:28 | comment | added | Overflowian | So $-\Delta-\lambda_1$ is positive semi-definite, does it still have a unique extension? | |
| Apr 20, 2023 at 12:24 | comment | added | MaoWao | I took $\lambda_1$ to mean the bottom of the spectrum of $-\Delta$ (which is actually contained in the spectrum because it is closed). Of course, if $\lambda_1$ is embedded in the spectrum, this is no longer true -- then you can have approximate eigenfunctions of the bottom of the spectrum for which $e^{t(\Delta+\lambda_1)}u_0$ blows up in $L^2$ norm. | |
| Apr 20, 2023 at 12:05 | comment | added | Overflowian | I'm a bit confused by signs here, you seems to say that the spectrum of $-\Delta $ satisfies $\sigma(-\Delta)\subset (\lambda_1,+\infty)$. So is the definition of $\lambda_1 $ given by $\inf \sigma(-\Delta)$?. I thought that it was the smallest positive eigenvalue and in particular it should be contained in the spectrum. | |
| Apr 20, 2023 at 9:43 | history | edited | MaoWao | CC BY-SA 4.0 |
added 68 characters in body
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| Apr 20, 2023 at 6:38 | history | answered | MaoWao | CC BY-SA 4.0 |